L10n105

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L10n104.gif

L10n104

L10n106.gif

L10n106

Contents

L10n105.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n105 at Knotilus!


Link Presentations

[edit Notes on L10n105's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,16,3,15 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X17,1,18,4
Gauss code {1, -4, -3, 10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {4, 5, -10, 9, -8, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n105 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(4) t(3)^2-t(4)^2 t(3)-t(1) t(2) t(3)-t(1) t(4) t(3)+t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+t(4) t(3)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial -\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}+\sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial 2 a^5 z^{-1} +a^5 z^{-3} -4 z a^3-7 a^3 z^{-1} -3 a^3 z^{-3} +2 z^3 a+6 z a+8 a z^{-1} +3 a z^{-3} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3} (db)
Kauffman polynomial 3 a^5 z^3-a^5 z^{-3} -8 a^5 z+5 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+10 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^7-5 a^3 z^5+15 a^3 z^3-3 a^3 z^{-3} -19 a^3 z+12 a^3 z^{-1} +2 a^2 z^6-8 a^2 z^4+20 a^2 z^2+6 a^2 z^{-2} -19 a^2+a z^7-5 a z^5+15 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -19 a z-8 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +z^6-4 z^4+10 z^2+3 z^{-2} -10 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-1012χ
4      22
2     121
0    3  3
-2   13  2
-4  31   2
-6  3    3
-821     1
-102      2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-4 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2} {\mathbb Z}^{2}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L10n104.gif

L10n104

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L10n106